Question

Inverse function of f(x) = log₄ (x-5) 2?

Answer 1

To find the inverse function of f(x) = log₄ (x-5)^2, convert the function to its exponential form, solve for x, and remember that exponential and logarithmic functions are inverses. The inverse function is f⁻¹(y) = 4√y + 5.

Step-by-step explanation:

The student is asking how to find the inverse function of f(x) = log₄ (x-5)^2. To find the inverse of a log function, we remember that exponential and logarithmic functions are inverses. Following this principle, we convert the log equation to its exponential form to find the inverse.

Here are the steps:

1. First, write the given function as an equation: y = log₄ (x-5)^2.
2. Convert the log equation to its exponential form: 4^y = (x-5)^2.
3. Solve for x: Take the square root of both sides to get ±4√y = x - 5. Remember that we are looking for the positive solution since x-5 was squared; this means we will use the positive root only.
4. Then add 5 to both sides to isolate x: x = 4√y + 5.
5. Therefore, the inverse function is f⁻¹(y) = 4√y + 5.

This process utilizes the basic rule that applying logarithmic and exponential functions sequentially will cancel each other out, which is clearly demonstrated in the step-by-step transformation of the given logarithmic function into its inverse exponential form.